Mixture Volatility Models and Exotic Derivative Pricing

# Why is the LMM with mixture dynamics (Brigo & Mercurio) inconsistent for the pricing of exotics?


# Why is the LMM with mixture dynamics (Brigo & Mercurio) inconsistent for the pricing of exotics?












I am reading about the LMM with lognormal-mixture dynamics. Consider the following dynamics for the forward rate $F_{i}(t)$ fixing at $T_{i-1}$ and paying at $T_i$: \begin{align} dF_{i}(t) = (F_i (t) +\gamma) \sigma_i^I dt \ , \end{align} where $\gamma$ is a constant shift and $\sigma_i^I$ is a random variable for the volatility drawn at time $0^+$ from a discrete or continuous distribution (the superscript $I$ denotes the state). These dynamics have the convenience that the price of a European call/put simply is the expectation of Black-Scholes prices over all possible scenarios for the volatility, and one could quite easily calibrate the states on the market smile of the caplet.

In an article by Piterbarg, "Mixture of Models: A Simple Recipe for a... Hangover?", it is argued that such a dynamics cannot be used for the pricing of exotics (one should for example first transform the dynamics into a local vol function).

However, I fail to understand his argument, especially on page 4. The author takes as an example a simple compounded put option with two states {1, 2} for the volatility. He then argues that when time passes by and if the market doesn't change, the continuation value $H(S)$ at time $T_1$ (first exercise date) will be calculated under the same mixture as follows: \begin{align} H(S) = p_1 E^1_{T_1}\{(K_2 - S_{T_2})^+ |S_{T_1} = S \} + p_2 E^2_{T_1}\{(K_2 - S_{T_2})^+ |S_{T_1} = S \} \ . \end{align} First of all, what I don't understand is that when a book of exotics is marked-to-market on a daily basis, even if the market doesn't change significantly, a new calibration procedure could lead to different calibrated states simply because the time to maturity changes (i.e. we get closer to $T_1$ and $T_2$ in the example above).

Does anybody has a clear explanation (either intuitively or mathematically) on why such a models can or can't be used in practice? Intuitively I would think that such a model would perform at least better than a simple LMM (only one state with probability 1) when it comes to hedging the smile. I understand that theoretically the model isn't very realistic, but I am looking at the problem from a practitioner's point of view. Any insights on the problematic would be useful.

## Answer by jaehyukchoi49 (score 2, accepted)

https://quant.stackexchange.com/a/71210

My impression is that Piterbarg criticizes the mixture approach because it is not consistent with the stochastic volatility (SV) world. In the SV world, instanteneous volatility (or variance) continuously evolves over time, although it's not observable. In the mixture model, volatility discountinuously evolves immediately after $t=0$, which is not an ideal picture in the SV world. In short, if you're religious to the SV world, then, the mixture model is nonsense.

However, there's an approach to view volatility an uncertain quantity (it's unobservable) and to assume it as a distribution. There's a technique called `randomization`. For example, see Jacquier & Shi (2019) where the initial variance of the Heston model is assumed as a random variable. In this respect, the mixture approach makes some sense. IMHO, I think the mixture model may not be perfect but is better than the simple (fixed volatility) model.

References:

- Jacquier A, Shi F (2019) The Randomized Heston Model. SIAM J Finan Math 10:89–129. https://doi.org/10.1137/18M1166420. Download

## Answer by solid (score 0)

https://quant.stackexchange.com/a/81459

From Piterbarg, "Mixture of Models: A Simple Recipe for a... Hangover?"

> $$ V_{\text{MM}} = \sum_{i=1}^N p_i V(S_i), \tag{1.1} $$ If one is only interested in valuing European options, using mixture models is appropriate, as the (multiple possible) dynamics have no bearing on their values, and the formula (1.1) can be used. For anything more complex, one has to do two things. First, fully specify the evolution of all the state variables in the model through time. Second — do not use the formula (1.1). This has been the approach of Brigo and Mercurio. Brigo and Mercurio start with a model for a stock price such that all one-dimensional distributions in the model are mixtures of lognormals. They, however, do not propose to use a weighted average of Black-Scholes models to value all derivatives. Instead, they derive a local volatility model for the stock price consistent with the assumption of lognormal distribution mixture. A local volatility model is, of course, fully self-consistent. It is the latter model that is then applied to price all derivatives of interest.

Piterbarg is not criticizing the idea of a mixture-model, but the idea of using a weighted average of derivative security prices computed using different “simple” models (the so-called “mixture of models”, or “ensemble of models”, approach) as a simple way to add stochastic volatility to virtually any model - and pricing a derivative which depends on the dynamics of volatility uncertainty.

Piterbarg's paper criticize the idea of this non-dynamic stochastic volatility model which only specifies what is going to happen to those volatility scenarios at a fixed time in the future, and not what happens in between now ($t_0$) and then ($T$) - which is reflected in the example provided, where the probabilities (weights) of the volatility scenarios remain constant over time. This approach is unsuitable for valuing path-dependent derivatives (do depend on the path between now and maturity), as it does not account for how volatility uncertainty evolves.

As it is mentioned above, in the lognormal-mixture model (Brigo, Mercurio) it is specified the evolution of all the state variables in the model through time - and, there is no claim of using the formula (1.1) for the pricing of all derivatives. To use a mixture model to price path-dependent products, one must specify the mixture model dynamics, particularly how probabilities (weights) of different volatility scenarios evolve over time. This is done in the lognormal mixture model, where the risk-neutral density of the asset is a mixture of lognormal densities, and an analytical diffusion coefficient is specified. The square of the local volatility $\nu(t,y)$ is a weighted average of the squared basic volatilities $\sigma_1^2(t), \dots, \sigma_N^2(t)$, with weights as functions of the marginal lognormal densities:

$$ \nu^2(t, y) = \sum_{i=1}^N \Lambda_i(t, y) \sigma_i^2(t) = \frac{\sum_{i=1}^N \lambda_i \sigma_i^2 p_i(S)}{\sum_{i=1}^N \lambda_i p_i(S)}. $$